Modelling and explaining a mathematical concept

With thanks to Nick Hart, Deputy Headteacher at Penn Wood Primary School in Slough, for this really interesting article on modelling mathematical concepts.

Sound understanding of mathematical concepts is the holy grail of maths teaching, but it is fraught with difficulty. Even with good subject knowledge and a flawlessly delivered explanation, followed by some well-designed work to do, some children still misconceive. If we don’t get our explanations and tasks right, there is a huge chance that misconceptions will engender and fester. Here are 5 phases of teacher modelling to set children up to succeed.

1.Show examples

By showing two more examples of a concept, we can highlight what it is about each example that is important and what is unimportant. For example, if we’re teaching children what a square is, we could show the following examples:

These examples show that squares have four equal sides but that size does not matter – squares can be big or small. Inevitably, there’ll be other characteristics that we’d need to draw children’s attention to:

Here, we’re showing children that squares have four equal sides but that orientation does not matter (and that the second shape is not a diamond!). The models and images that we choose help to focus explanations. In some ways, explaining a concept is harder than explaining a process, particularly because, as teachers, such concepts are so ingrained in us that verbalising the defining characteristics can be difficult. The concept of a square is very simple, but in other areas of maths it can be all too easy to answer a ‘Why …?’ question with ‘Because it just is!’

2. Show non-examples

In order to strengthen children’s understanding of what something is, it is useful to point out what is is not. Doing so helps to reinforce the characteristics pointed out in the first phase and exemplify more important characteristics. Indeed, it is sometimes easier to point out a defining characteristic when it is not there. Sticking with the square example:

3. Clarify the characteristics

Having shown children some examples and non-examples one at a time to keep cognitive load bearable, they can be presented side by side now in order to give children another interaction with the content and to generate success criteria to be used in this lesson and beyond.

If the first two phases were well rehearsed teacher explanations of what squares are or are not, then this phase is children elaborating on those same prompts, using talking partners to use the right mathematical understanding and to practise noticing the key characteristics. After a brief discussion, we can succinctly compile the criteria for what makes a square a square:

4 sides

Equal sides

4 right angles

Can be any size

Can be any orientation

These success criteria become a checklist for children to work with when they start working with shapes of their own.

4. Shared mathematical thinking

In the first three phases, we’ve ensured that children have access to the essential knowledge to now work with the concept – in this case squares – whether that is partly internalised or they must use the working wall until it is internalised. The next stage is to get children to think mathematically using that knowledge. A simple sorting activity could be the starting point. Using a range of shapes (concrete or pictorial), we can guide children into analysing them, consulting the criteria and deciding whether they are squares or not. The teacher’s questioning here would be around drawing children’s attention to observable or absent characteristics of the shape and then prompting them to make a decision on whether it is or is not a square.

An alternative task to sorting could be presenting shapes one at a time and asking: is this a square? After some modelling of well-crafted sentences that use the correct vocabulary, some shared speaking and writing could be done to help children create and refine mathematically accurate and efficient reasoning responses.

Another alternative is to set statements about the content and ask children whether they are always, sometimes, or never true. For example:

Squares have 4 equal sides – always true.

Squares are small – sometimes true.

Squares have an angle bigger than a right angle – never true.

Through a combination of explanations and drawings, children can prove whether each statement is always, sometimes or never true. The beauty of this task is that you can riddle the statements with likely misconceptions to really get children thinking and to reveal the depth of their understanding for swift intervention. This doesn’t always work - pretty soon there’ll be no other worthwhile statements to be made about squares; but for some other concepts, there is more scope for more statements to be made.

5. Independent mathematical thinking

Whatever thinking that we have modelled and collaborated on should be the kind of thinking that children have a go at for an extended period of time independently. With the scaffold of whole-class work gone, it is here that misconceptions will arise that we can pick up on in focus groups or through marking. Of course, some children will need work scaffolded and some will have grasped the concept quickly, and these differences will need to be catered for. We mustn’t fall into the trap of thinking that one lesson on a concept will cut it either. The initial phases of the lesson will have highlighted important knowledge that needs to become autonomously recalled at will, and so from that point on there needs to be multiple interactions and regular low-stakes testing, particularly for our most disadvantaged.